Elements of the Computer
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Today's computer is the culmination of thousands of years of human mathematical thought. This section describes that process.
Contents |
Numbers
Although digital files are a modern phenomena, the concept of using numbers to describe things is much, much older. Counting, and its companion, numbering, are as old as humankind itself. Animal bones that date back to at least 15,000 BCE. have been found with notches in them, which are thought to be tally marks. These marks directly signify the things they represent: one mark for each bison, for example. Numbers, on the other hand, are an abstraction of these tally marks. Compare the number 5 to the five lines carved on a wall or a tree. Numbers are a precursor of the modern process of digitization: the description of concrete reality by abstract number.
Early numbering and calculating systems, such as those developed in Mesopotamia in the fourth millennium BCE, depended on direct physical analogs: one pebble for one sheep, two pebbles for two sheep. By the third millennium BCE, the Egyptians and the Sumerians had developed the written digit. Once numbers had entered the abstract world of writing and were independent of the objects they represented, they could be manipulated in new ways. Concepts of manipulating numbers developed into addition, subtraction, and so forth. In order to count, we must understand that things exist discretely from one another: an individual rock is complete in and of itself, and separate from the other rocks in a pile. Digital systems, those that rely on numbers or digits, are said to be discrete, such as whole numbers or integers. Analog systems are continuous and have no breaks or steps—just think of thermometers: an analog thermometer has a band of mercury that extends itself continuously as the temperature rises, whereas a digital thermometer shows a changing series of numbers.
Digital v. Analog
Signals can be represented in digital or analog form, terms that refer to the methods used for encoding and storage. Analog refers to a process through which a physical trace of the signal is made. In the digital method, the signal is reduced to digits or numbers, which represent aspects of the signal or information. Digital encoding is infinitely reproducible: copies can be made without any loss of quality because the process entails nothing more than the transcription of a set of numbers. Analog duplications lose clarity—it is difficult to duplicate a physical object perfectly. An analog representation of a signal is made by allowing the signal to act directly on a physical substrate, such as light on photographic film. The quality of an analog image is determined by the sensitivity of the film and the quality of the optics of the camera—their ability to create a faithful duplicate. It was Thomas Edison (American inventor and businessman, 1847-1931), in the late nineteenth century, who developed the analog process used to cut tracks in a record album. He discovered that sound caused a diaphragm in a microphone to vibrate, which in turn moved a stylus that physically cut into the surface of a rotating wax cylinder. Sound unfolds through time, and with the phonograph, the time of the sound is mapped onto the space of the cylinder through its rotation. Through the stylus, sound acts physically and continuously on the surface of the album, creating a physical analog. Playback is the reverse process: a stylus is placed in the groove of a record and then vibrates according to its irregularities. The vibrations of the stylus are physically carried to a speaker, which produces sound. The analog recording is very deeply tied to its physical existence; when its medium changes (for example, from album to tape), the sound quality changes.
In a digital music file such as an MP3, sound is represented by binary numbers. The sound is first fed into a computer, where it is sampled and converted into numbers that denote various elements of the sound, such as frequency, timbre, and amplitude. In order to play a digital file, the computer or CD player must read the numbers, then interpret them and translate them back into analog signal to be carried to the speakers. The medium that contains the digital information is irrelevant—it could be a disc, a tape, or even paper. Both analog and digital methods must in some way create a souvenir of the event, a trace of the sound. Whereas analog record albums were stored in libraries and public or private collections, the process of digitization has moved music into the virtual network, accessible to everyone.
Decimal and Binary
Our decimal counting system (base 10, using ten discrete symbols, the numbers zero through nine) is very probably a remnant of counting on one's fingers. If you add your two feet, you can count to twelve, or a dozen. Our abstract, notational system of numbers is tied to the limits of our extremities. But this is not universal. The Aborigines who live in the Torres Strait near Australia use a base two (binary) system, the Sumerians used a base 60, and the Mayans used a base 20. Shao Yung (Chinese philosopher, 1011-1077) proposed a mathematical system based on binary numbers that was a direct influence on Gottfried Wilhelm Leibniz's (German philosopher and mathematician, 1646-1716) use of the binary system. In fact, Leibniz preferred binary to decimal for much of his writing. Some early computers - notably the ENIAC of the mid-1940s—used decimal numbers, stored in complicated devices called decade counters, but their designers quickly realized several advantages to using binary numbers. A binary numbering system, composed of nothing but zeros and ones, is easier to store in an electronic circuit, as either the presence or absence of current in a circuit. More importantly, early computer designers saw that binary numbers shared many properties with the formal logic on which computer programs are based. The EDVAC, completed shortly after ENIAC, was able to store both its data and program in binary digital form.
Algorithms
If numbers are an abstract concept that allows us to understand the world through counting, then mathematics is an abstract set of rules that allows us to manipulate numbers and so better understand and regulate the world. Euclid (Greek philosopher and mathematician, active circa 300 BCE) advanced the field of geometry (literally the measurement of land) in his third-century BCE work, The Elements (a compilation of previous thinking about algebra and geometry), as a part of a rigorous system of philosophical thought.
The Greek contributions to the field of mathematics were the notions that mathematics should be general and that mathematical statements should be proven. These proofs laid the foundation for the algorithm, building on the mathematics of the Mesopotamians, who had developed procedures for finding whole numbers.
Euclid's Elements contained methods for finding the common divisor of two numbers. This was an algorithm, a set of rules for finding the solution to a problem (a computer program is a type of algorithm, one translated into a code the computer can understand, usually through a programming language). The term "algorithm" comes from the Latin translation, algorismi, of the name of the Islamic mathematician Muhammed ibn Musa al-Kwarizmi (Persian mathematician, circa 780-850). In the ninth century, he was one of many thinkers who worked in the House of Wisdom in Baghdad and was exposed to translations of Greek works. He wrote two books— The Hindu Calculation, which introduced Hindu notions of arithmetic to the Arab world, and the Book of Restoring and Balancing, which introduced algebra to the West, and from whose title the word "algebra" is derived (from the Arabic al-jabar, restoring).
Wikipedia defines an algorithm as "a finite list of well-defined instructions for accomplishing some task that, given an initial state, will terminate in a defined end-state." This mathematical concept is the basis for modern computer programming.
Punched Cards
To understand the modern digital file, one must accept that the image or sound (and so forth) no longer consists of a physical trace of the object represented; instead, it is a virtual, numerical representation of that object. Following this definition, the first digital image can be traced back to 1801, when Jean-Marie Jacquard (French weaver and inventor, 1752-1834) demonstrated an automated loom whose actions were directed by a program stored on punched cards, a precursor of the computers of the 1960s, which stored their programs on cards of this nature. These cards contained information encoded on them by a series of holes punched in specific locations, like those in the rolls of a player piano. The information on the cards, the presence or absence of a hole in a particular spot, would determine the movements of the loom to weave together the threads, and thus the pattern of the cloth that the loom produced. Jacquard also produced a self-portrait on the loom, encoded on 10,000 cards used to weave the portrait. We can consider this image digital insofar as it represented the image with discrete units, and required the intervention of an interpretive machine—the loom—to create the image from stored data. Charles Babbage (English mathematician and philosopher, 1791-1871) was directly influenced by Jacquard's use of cards, and it is said that he had a portrait of Jacquard hanging on the wall of his office.
